If two vertices of an equilateral triangle are (1, -1) and (-root 3, -root 3), find the coordinates of the - Maths - Mathematical Reasoning

Question

If two vertices of an equilateral triangle are (1, -1) and (-root
3, -root 3), find the coordinates of the third vertex

Zuhaib Ul Zamann · Accepted Answer

Dear student,
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Let A≡(1,-1) and B≡(-3,-3)
 let third vertex be  C≡(x,y)Slope of AB=-3+1-3-1=2-3Hence AB is at angle of tan-1(2-3) =15∘ with x-axisHence slope of AC is tan(15+120)=-cot(45)=-1 or tan(15-120)=cot(15)mAC=-1 or 2+3Also when slope of AC is -1  slope of BC=tan(60+15)=2+3and when slope of AC is 2+3 slope of BC=tan(15-60)=-1Hence mBC=2+3 or -1Case ImAC=-1  mBC=2+3equation of ACy+1=-1(x-1)y=-x   (I)equation of BC isy+3=(2+3)(x+3)   IIsolving I and II we getx=-1 ,y=1i
e C≡(-1,1)Case IImAC=2+3   mBC=-1equation of AC=y+1=(2+3)(x-1)    IIIequation of BCy+3=-1(x+3)   IVsolving III and IV we getx=2-3,y=-2-3So C≡(2-3,-2-3)Regards
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If two vertices of an equilateral triangle are (1, -1) and (-root 3, -root 3), find the coordinates of the third vertex.